Linear Algebra – How to Relate Frobenius Norm and Trace of a Matrix

Question

I came across an expression relating the Frobenius Norm with Trace as follows :

$$\|UU^T\hat X\|^2_F = tr((UU^T\hat X)^T(UU^T\hat X))$$

Could someone explain briefly the equivalence ?
THanks

Answer 1

For any matrix $M$, we have $$ \|M\|_F^2 = \operatorname{tr}(M^TM). $$ In particular, note that the $j,j$ entry of $M^TM$ is equal to $\sum_{i=1}^n m_{ij}^2$, so that the sum of the diagonal entries (the trace) is indeed equal to the sum of the squares of all entries of $M$.